Elena Comparini

A one-dimensional Bingham flow

A Bingham fluid is a non-Newtonian fluid, which behaves like a rigid body when the shear stress t is less than a threshold value tO, while it behaves like a viscous fluid when the stress exceeds rO, and for which the relationship between stress and strain rate rr is linear, where q is the viscosity. A behaviour like that of Bingham fluids can be observed in the materials used in the industrial processes for fabrication of ceramics and paper, or in many types of slurries. A one-dimensional evolution model describing the flow of a Bingham fluid was formulated in [l], where the equations governing the flow are the Navier-Stokes equation in the viscous region (the field equation) and the equation of motion for the rigid part (playing the role of a free boundary condition). The boundary of the rigid core is the free boundary, where we have T = rO, i.e., zero strain rate. The problem was formulated in a weak form by Duvaut and Lions [2] as a variational inequality. They studied an initial value problem and obtained weak solutions, whose regularity was proved in the two-dimen- sional case. In that case, they also proved the convergence of the solution of the problem for Bingham fluids to the one for Newtonian fluids, when the shear stress tends to the threshold value. The existence and uniqueness in a certain function class of a strong solu- tion to the variational inequality associated to the flow in the plane and in a bounded three-dimensional domain have been established in [3,4]. The steady state motion of a Bingham fluid in cylindrical pipes subject to a given constant pressure gradient has been investigated in [IS, 6,7]. i27

Shock Propagation in a One-Dimensional Flow Through Deformable Porous Media

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution possibly in the case of onset of singularities, which can be interpreted as a local collapse of the medium. In particular we analyze the case in which the boundary pressure has a piecewise constant derivative.

SHOCK PROPAGATION IN A FLOW THROUGH DEFORMABLE POROUS MEDIA: ASYMPTOTIC BEHAVIOUR AS THE POROSITY APPROXIMATES A CONSTANT

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered as functions of the flux intensity. We prove that if one approximates the porosity with a constant then the solution of the hyperbolic problem converges to the classical continuous Green–Ampt solution, also in the presence of shocks. In general, however, the shocks remain present in any approximating solution.

Convergence to the psuedo-steady-state approximation for the unreacted core model

We prove that the solution of a parabolic free boundary problem, arising from a model for some isothermal equimolal non-cathalytic reactions between a fluid and a solid (e.g. oxidization), converges to the solution of the pseudo-steady-state approximation.

Travelling wave solutions of a reaction—infiltration problem and a related free boundary problem

We consider travelling wave solutions of a reaction–diffusion system arising in a model for infiltration with changing porosity due to reaction. We show that the travelling wave solution exists, and is unique modulo translations. A small parameter e appears in this problem. The formal limit as e → 0 is a free boundary problem. We show that the solution for e > 0 tends, in a suitable norm, to the solution of the formal limit.